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Distance traveled by random walkers before absorption in a random medium
David S Dean1, Clément Sire, Julien Sopik
1Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4, France. dean@irsamc.ups-tlse.fr
Summary
We determined the penetration length (l) for random walkers in media with absorbers. The length depends on absorber density (rho) and correlation length (xi) in dimensions D > 2.
Area of Science:
- Physics
- Statistical Mechanics
- Physical Chemistry
Background:
- Random walkers are fundamental to modeling diffusion processes.
- Understanding absorber effects is crucial for various physical phenomena.
- Previous studies lacked a unified approach across dimensions and absorber types.
Purpose of the Study:
- To determine the penetration length (l) of random walkers in media with absorbers.
- To analyze the influence of absorber density (rho) and correlation length (xi).
- To investigate behavior in different dimensions (D) and under long-range correlations.
Main Methods:
- Mean-field renormalization group applied to lattice and continuum models.
- Analysis across all dimensions (D).
- Illustration with exactly solvable toy models and numerical simulations (directed percolation).
Main Results:
- For D > 2 homogeneous systems, penetration length l ~ max(xi, rho^(-1/2)).
- Behavior in D = 1 and D = 2 dimensions was analyzed.
- Temporal decay of unabsorbed walkers estimated in the presence of long-range correlations.
Conclusions:
- The study provides a comprehensive framework for random walker penetration length.
- Results have implications for diffusion-limited aggregation (DLA).
- A more effective method for measuring penetration length in DLA clusters is proposed.
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